in·du·rate Pronunciation[v. in-doo-reyt, -dyoo-; adj. in-doo-rit, -dyoo-; in-door-it, -dyoor-] verb, -rat·ed, -rat·ing, adjective–verb (used with object)1. to make hard; harden, as rock, tissue, etc.: Cold indurates the soil.2. to make callous, stubborn, or unfeeling: transgressions that indurate the heart.3. to inure; accustom: to indurate oneself to privation and suffering.4. to make enduring; confirm; establish: to indurate custom through practice.–verb (used without object)5. to become hard; harden.6. to become established or confirmed.–adjective7. hardened; unfeeling; callous; inured.

Hard to say.
It seems that as an adjective, they are synonyms. Obdurate does not have a verb form, as far as I can tell.

induratev. in·du·rat·ed, in·du·rat·ing, in·du·rates v.tr.1. To make hard; harden: soil that had been indurated by extremes of climate.2. To inure, as to hardship or ridicule.3. To make callous or obdurate: "It is the curse of revolutionary calamities to indurate the heart" Helen Maria Williams.v.intr.1. To grow hard; harden.2. To become firmly fixed or established.adj. (nd-rt, -dy-)Hardened; obstinate; unfeeling.

"Time is nature's way of keeping everything from happening all at once. Lately it hasn't been working."Anonymous

Obdurate was a GWotD here, with Doc's usual entertaining etymology. Indurate would make a fine add for the future.

As we generally use a long A to denote the verb form and short for adjective, it looks to me that they actually would sound different. It appears indurate is most commonly used as a verb, hence a long A, where obdurate's an adjective.

[indurate gets a spell flag for me too, but then so does obdurate's. These things are obdurate.]

Here is a contextual example from A Distant Mirror by Barbara Tuchman. The topic is possible ways of resolving the schism between Pope Clement in Avignon and Pope Boniface in Rome: "The Three Ways now proposed were, first, mutual abdication; second, if both popes continued obdurate, arbitration by a selected group; third, a General Council of the Church."

This usage seems consistent with both definition 1 and 2 above, and, given that the period of the pope and anti pope was the latter half of the 14th century and the first half of the 15th, also consistent with its entry into the language ("1400 - 50; late ME obdurat").